Three statements with different mathematical types

The following objects must not be identified:

1. local identities for field insertions and current smears;

2. a strongly continuous unitary group representation, together with covariance of the bounded net;

3. a construction of the spacetime metric from other data.

A global implementation, if proposed, must specify its group or covering group, region action, Hilbert-space representation, domains of generators, and covariance:

whenever the geometric and algebraic expression is defined. If $gO$ leaves the Minkowski chart, an appropriate extension of the net must be supplied.

A local Ward certificate below does not supply these objects.